Response property of a factional linear system under the base excitation

We investigate the response property of a linear system that is excited by the base excitation. The linear system contains the ordinary damping or the fractional-order damping. In our studies, the base excitation is in the harmonic form or in the general periodic form. When the base excitation is in the harmonic form, we obtain the dynamic transfer coefficient by the undetermined coefficient method. When the base excitation is in the general periodic form, we first expand the excitation into the Fourier series, then, according to the linear superposition principle, we obtain the dynamic transfer coefficient that is induced by each harmonic component in the excitation. By expanding the general periodic excitation into the Fourier series, we can solve the non-differentiable problem that is induced by the periodic base excitation for the numerical calculations. Based on the Grnwald-Letnikov definition, the discretization formula for the fractional-order system is obtained explicitly. The analytical results are in good agreement with the numerical simulations, which verifies the validity of the analytical results. Both the analytical and the numerical results show that the dynamic transfer coefficient depends on the fractional-order of the damping closely. The dynamic transfer coefficient can be controlled by tuning the value of the fractional-order. For the general periodic excitation, when the frequency is fixed, the dynamic transfer coefficient that is induced by the high-order harmonic component may be stronger than that induced by the low-order harmonic component in the base excitation. Hence, the effect of the high-order harmonic component in the excitation cannot be ignored although its amplitude is small. Further, when the base excitation is in the full sine form, or the square form, or the triangular form, the response property of the system can be described by center frequency, resonance peak, cutoff frequency, and the filter bandwidth. For a fixed fractional-order, the center frequencies of each order corresponding to the response, obtained by the three kinds of the periodic base excitations mentioned above, are identical. However, the corresponding resonance peaks are different. The resonance peak and the filter bandwidth are both maximal when the base excitation is in the square form. The resonance peak and the filter bandwidth are both minimal when the base excitation is in the triangular form. We believe that our results are useful for solving the vibration problem in the engineering field such as the vibration isolation and the vibration control.